Thermodynamics of Solids Under Pressure

Home , Materials physics, Thermal pressure

CHAPTER 1 Thermodynamics of Solids under Pressure AlbertoCopyrighted Otero de la Roza MALTA Consolider Team and National Institute for Nanotechnology, National Research Council of Canada, Edmonton, Canada Víctor Luaña MALTA Consolider Team and Departamento de Química Física y Analítica, Universidad de Oviedo, Oviedo, Spain Manuel Flórez MALTA Consolider Team and Departamento de Química Física y Analítica, Universidad de Oviedo, Oviedo, Spain Material

CONTENTS

1.1 Introduction to High-Pressure Science ...... 3 1.2 Thermodynamics of Solids under Pressure- ...... 6 1.2.1 Basic Thermodynamics ...... Taylor 6 1.2.2 Principle of Minimum Energy ...... 14 1.2.3 Hydrostatic Pressure and Thermal Pressure ...... 15 1.2.4 Phase Equilibria ...... 17 1.3 Equations of State ...... 19 1.3.1 Birch–Murnaghan Family ...... & 20 1.3.2 Polynomial Strain Families ...... 21 1.4 Lattice Vibrations and Thermal Models ...... Francis 23 1.4.1 Lattice Vibrations in Harmonic Approximation ...... 23 1.4.2 Static Approximation ...... 26 1.4.3 Quasiharmonic Approximation ...... 29 1.4.4 Approximate Thermal Models ...... 32 1.4.5 Anharmonicity and Other Approaches ...... 40 1.5 Conclusions ...... 42 Bibliography ...... 44

1.1 INTRODUCTION TO HIGH-PRESSURE SCIENCE

igh pressure research [1] is a ﬁeld of enormous relevance both for its sci- H entiﬁc interest and for its industrial and technological applications. Many

3 4 An Introduction to High-Pressure Science and Technology

materials undergo fascinating changes in their physical and chemical characteristics when subjected to extreme pressure. This behavior is caused by the involvement in bonding of electrons that would otherwise not be chemically active under zero-pressure conditions (the difference between zero pressure and atmospheric pressure is negligible from the point of view of its effect on materials properties, and we will use both concepts interchangeably). By and large, current chemical knowledge and the traditional rules for valence electrons are at a loss to explain most of the changes induced when materials are com- pressed, which makes chemical bonding under pressure an exciting research topicCopyrighted that has received much attention in recent years [1] (see Chapter 5). The study of pressure effects can be approached in two mostly complemen- tary ways: experimentally and computationally. In both cases, the basic object under study is the change in crystal structure a material undergoes when a given pressure and temperature are applied as shown in its thermodynamic pressure-temperature phase diagram. Experimentally, the application of temperature is relatively straightforward, but imposing high pressure on a sample requires specialized techniques that have been under development for the past 80 years. The field of experimentalMaterial high-pressure physics was pioneered by Percy Bridgman, who received the Nobel Prize for his efforts in 1946 [2]. There are two main experimental high-pressure techniques: dynamic compression methods based on shock waves, and static compression methods which make use of pressure cells. In a shock-wave compression experiment [3], a strong shock is applied to the sample and- the propagation velocities of the shock wave inside the material are measured.Taylor Very high pressures (up to 500– 1000 GPa) and temperatures (tens of thousands of K) can be attained with this technique. Static compression techniques are applied using pressure cells, notably diamond anvil cells [4]. These methods are more accurate than dynamic techniques, but they are also limited by the pressure& scale, with the mechanical resistance of a diamond (slightly above 300 GPa)Francis being the ulti- mate upper pressure limit for the technique. High temperatures, in the range

Downloaded by [Taylor and Francis/CRC Press] at 12:53 06 April 2016 of thousands of K, can be attained by laser heating, and diamond anvil cells can be coupled to other techniques: spectroscopic (infrared, Raman, x-ray) and optical. Experimental high-pressure techniques are explored in Part II of this book. From the computational point of view, in the framework of density func- tional theory (DFT), the application of pressure is relatively straightforward: one simply compresses the unit cell and calculates the applied pressure either analytically from the self-consistent one-electron states or from the volume derivative of the calculated total energy. The latter requires fitting an equation of state. Conversely, temperature effects are difficult to model because they involve collective atomic vibrations—called phonons—propagating throughout Thermodynamics of Solids under Pressure 5

a crystal (see Chapter 3). The simplest computational approach to model the effects of temperature is to calculate the electronic energy, assuming the nuclear and electronic motions are decoupled (the adiabatic approximation), and then obtaining the phonon frequencies using the derivatives of said electronic energy with respect to the atomic movements. The phonon frequencies can be used in the harmonic approximation to calculate thermodynamic properties and phase stabilities at arbitrary pressure and temperature. Computational techniques for high-pressure studies are explored in this and the subsequent chapters in Part I. DespiteCopyrighted the difficulties in modeling temperature effects, theoretical approaches are necessary in cases for which experiments are either not feasible or difficult to interpret. An interesting example of synergy between computational and experimental approaches occurs in geophysics. Because the interior of the Earth is inaccessible, information about its composition and structure can only be inferred through indirect means [5, 6], primarily via seismic velocities measured during earthquakes. Calculated data (elastic moduli, longitu- dinal and shear velocities, thermal expansion and conduction, heat capacities, Grüneisen parameters, transitionMaterial pressures, Clapeyron slopes, etc.) are very helpful in the interpretation of experimental data and in the construction of models of the Earth’s interior [7, 8]. The geophysical and astrophysical applications of high-pressure science are presented in Chapters 15 and 16, respectively. High-pressure research also has very- relevant technological applications (Chapter 14). Materials that are thermodynamicallyTaylor stable at high pressure and temperature are often metastable at zero pressure and room temperature, so it is possible to access new material phases with unique properties by the application of appropriate pressure and temperature conditions. The application of high pressure to biological samples is also of technological& relevance because it is known that the microorganism activity is diminishedFrancis or canceled by application of high pressures, a process called pascalization. In contrast to

Downloaded by [Taylor and Francis/CRC Press] at 12:53 06 April 2016 the preceding applications, the pressures exerted in this case are in the order of the tenths of a GPa, much smaller than in previous examples. Pascalization can be used to increase the shelf lives of perishable foodstuffs: juice, fish, meat, dairy products, etc. This has been one of the research lines of MALTA [9–12], and is explored in Chapters 12 and 13. To summarize, high pressure is a very active field of research and one whose ramifications affect many scientific and technological fields, from astro- physics and geophysics to materials physics and the food industry. From the fundamental view, the behavior of materials at high pressure is still poorly un- derstood, and the usual textbook chemistry rules that apply to zero-pressure chemistry are essentially useless when molecules and materials are subjected 6 An Introduction to High-Pressure Science and Technology

to extreme compression. This makes high-pressure research an interesting and virtually unexplored ﬁeld of study. In the remainder of the present chapter, we give an overview of the basic thermodynamic principles and mathematical tools that are needed to master the rest of the book. Namely, we review the thermodynamics fundamentals (Section 1.2), the equations of state used to describe the responses of materials to compression (Section 1.3), and the harmonic approximation, the basic theoretical framework in which temperature eﬀects are modeled computationally (Section 1.4).

1.2Copyrighted THERMODYNAMICS OF SOLIDS UNDER PRESSURE 1.2.1 Basic Thermodynamics Hydrostatic pressure, together with its extensive associated variable volume, is a fundamental quantity covered in most thermodynamics textbooks [13–17]. Pressure can be deﬁned by considering the mechanical work exerted on a closed system (in which there is no matter exchange with the environment) through a change in its volume: Material δW = −PEdV . (1.1)

In this equation, PE is the pressure applied on the system or environmental pressure [18] (also, see the subsection on irreversible pressure-volume work and references therein in Chapter 2 of Reference 15). When both system and environment are in mechanical equilibrium- with each other, PE = p, with p representing the system’s pressure (constantTaylor throught the system). The definition of mechanical work is convenient to study, for instance, the behavior of gases and phase transitions in a closed system. Given that we are interested in the thermodynamic properties of solids under pressure, we start this chapter by presenting some fundamental quantities and the relations& between them that will be used in the rest of the book. Classical thermodynamics is a macroscopic theory basedFrancis on a few ax- iomatic principles derived from empirical observations. These fundamental Downloaded by [Taylor and Francis/CRC Press] at 12:53 06 April 2016 principles are called the laws of thermodynamics. The zeroth, first and second laws allow us to assert the existence of temperature T , internal energy U, and entropy S as state functions of the system (see below). The first law states that for any infinitesimal process in a closed system (at rest and in absence of external fields): dU = δQ + δW . (1.2) In this equation, dU is the infinitesimal internal energy change undergone by the system, δQ is the infinitesimal heat flow into the system, and δW is the infinitesimal work done on the system during the process. For a finite process, Q and W depend on the intermediate states through which the process evolves, Thermodynamics of Solids under Pressure 7

but ∆U depends only on the initial and the final states. This is symbolized in Equation (1.2) by using the exact differential notation (dU) for the internal energy but not for heat or work. Quantities like U that depend only on the state of the system are called state variables or state functions, and it is possible to define a function (whose changes can be empirically measured) that depends only on the variables that determine the state of the system, U(p, T, . . . ). As we show below, the same thermodynamic quantity can act as a variable (if it is independent) or as a function (if its value is given as a function of other independent variables). TheCopyrighted first law establishes the conservation of energy principle in an isolated system, in which there is no energy or matter exchange with the environment. That is, in an isolated system, the internal energy is a constant. When a closed system receives energy, either in the form of heat or work, its internal energy increases and the environment loses energy by the same amount. If one considers a cyclic process in a closed system, i.e., a process in which the initial and final states of the system are the same, the internal energy change of the system is zero: MaterialI dU = 0, (1.3)

which makes building a machine whose only effect is the constant generation of energy impossible. The second law of thermodynamics deals- with the difference between heat and work. It originates from the realizationTaylor that it is impossible to build a cyclic machine that transforms heat into work with 100 % efficiency. The second law, which was first proposed by Sadi Carnot through his analysis of steam engines, is therefore a principle of limited efficiency which, for a heat engine, is defined as the fraction of the heat input that is transformed into work. In the case in which heat is absorbed from a hot& reservoir and released into a cold reservoir, the maximum efficiency of a heat engineFrancis depends only on the temperature of the two reservoirs, and can only be attained if the engine

Downloaded by [Taylor and Francis/CRC Press] at 12:53 06 April 2016 works reversibly (infinitely slowly and excluding dissipative effects) through an infinite series of equilibrium steps [15]. For any real (irreversible) process happening at a finite velocity and including dissipative effects, the efficiency is always less than the reversible upper bound. The second law is formalized through the definition of entropy. Based on Carnot’s work, Clausius proved that for a reversible process in a closed system (whose temperature T is constant throughout), the quantity δQrev/T is an exact differential: I δQ rev = 0, (1.4) T 8 An Introduction to High-Pressure Science and Technology

or, in other words, we can deﬁne a function S (the entropy) as:

δQ dS = rev , (1.5) T which depends only on the state of the system. The difference between the entropy in states A and B, ∆S = SB − SA, can be calculated as the value obtained from Equation (1.5) in a hypothetical reversible process that goes from A to B. A corollary of the definition in Equation (1.5) is that the change in entropyCopyrighted of a closed system during a reversible adiabatic process is zero. The observation that it is not possible in practice to achieve the maximum thermal efficiency of an engine can be recast in terms of entropy. For an irreversible process in a closed system, Equation (1.5) becomes,

δQ dS > , (1.6) TE

with TE being the temperature of the environment. Therefore, the effect of any process on an adiabaticallyMaterial isolated system is to always increase its entropy. A corollary of this statement is that in an isolated system, thermodynamic equilibrium is attained once the entropy of the system is maximized, and no spontaneous processes within the system can occur henceforth. Note that TE = T , with T being the temperature of the system, only holds when both system and environment are in thermal- equilibrium with each other. As we shall see below, the maximum entropy statementTaylor of the second law can be recast into minimum energy principles that affect certain functions when the state of the system is given by their associated variables. For simplicity, let us now consider closed one-phase systems with pressure- volume work only undergoing a reversible process. Under& these conditions, the mechanical work effected on the system is given by δW = −FrancispdV and the heat absorbed by the system is given by δQ = T dS. The change in internal energy is thus given by the fundamental equation: Downloaded by [Taylor and Francis/CRC Press] at 12:53 06 April 2016

dU = T dS − pdV , (1.7)

which gives the expression of the exact differential dU in terms of two independent variables: entropy and volume. The internal energy U(S,V ) as a function of these variables permits the calculation of all the other thermodynamic properties of the system through its derivatives. U is called a thermodynamic potential and S and V are its natural variables. Entropy and volume are difficult variables to work with because, in general, isochoric and isoentropic conditions are not easy to achieve experimentally. Thermodynamic potentials that depend on other variables can be defined by Thermodynamics of Solids under Pressure 9

applying a Legendre transform to U(S,V ). In this way we deﬁne the enthalpy (H(S, p)), the Helmholtz free energy (F (T,V )), and the Gibbs free energy (G(T, p)) as:

H(S, p) = U + pV , (1.8) F (T,V ) = U − TS, (1.9) G(p, T ) = U + pV − TS = H − TS = F + pV . (1.10)

H, F , and G are the thermodynamic potentials for the indicated independent naturalCopyrighted variables. If a thermodynamic potential can be determined as a function of its natural variables, then all other thermodynamic properties for the system can be calculated by taking partial derivatives. For instance, if G is known as a function of T and p, the thermodynamic behavior of the system is completely determined. The U, H, F , and G thermodynamic potentials can be used to express criteria for spontaneous change and equilibrium when the corresponding natural variables are heldMaterial constant. For instance, G can only decrease during the approach to reaction equilibrium at constant T and p in a closed system doing pressure-volume work only, reaching a minimum at equilibrium. The fundamental relation for U (Equation (1.7)) can be combined with the definitions for the enthalpy and the free energies (Equations (1.8) to (1.10)) to find analogous fundamental equations for- the other thermodynamic potentials, valid for a reversible process: Taylor dH = T dS + V dp, (1.11) dF = −SdT − pdV , (1.12) dG = −SdT + V dp. & (1.13) In addition, a number of useful relations can be derived fromFrancis these equations. For instance, Equation (1.12) can be compared to the exact differential for

Downloaded by [Taylor and Francis/CRC Press] at 12:53 06 April 2016 F (T,V ): ∂F ∂F dF = dT + dV . (1.14) ∂T V ∂V T ∂F In this notation, the subscript in ∂T V is used to express that the remaining independent variables other than T for the F function (in this case, only V ) are held constant. Comparing Equation (1.14) and Equation (1.12), we arrive at: ∂F ∂F S = − , p = − . (1.15) ∂T V ∂V T 10 An Introduction to High-Pressure Science and Technology

Likewise, for the rest of the thermodynamic potentials,

∂U ∂U T = , p = − , (1.16) ∂S V ∂V S ∂H ∂H T = , V = , (1.17) ∂S p ∂p S ∂G ∂G S = − , V = . (1.18) ∂T p ∂p T Copyrighted The Maxwell relations can be obtained from the fundamental relations by calculating the mixed second derivatives of the thermodynamic potentials, which need to be the same regardless of the order in which they are taken. For instance, ∂2F ∂2F = (1.19) ∂T ∂V ∂V ∂T leads, using Equation 1.15,Material to: ∂p ∂S = . (1.20) ∂T V ∂V T Likewise, for the other state functions, one has:

∂p ∂T - = − Taylor(from U), (1.21) ∂S V ∂V S ∂V ∂T = (from H), (1.22) ∂S p ∂p S ∂V ∂S = − (from G). & (1.23) ∂T ∂p p T Francis These relations between thermodynamic potential derivatives are useful be- Downloaded by [Taylor and Francis/CRC Press] at 12:53 06 April 2016 cause, even though the potentials can not be measured directly, some of their variations correlate with quantities that are experimentally accessible. For instance, the volume derivative of the Helmholtz function (Equation (1.15)) gives the pressure as a function of volume and temperature, an object that is called the “volumetric equation of state”, or simply the “equation of state”, and which plays a key role in theoretical and computational studies of materials under pressure, as we shall see in the next sections. Other material properties and their relation to the corresponding thermodynamic potentials include the Thermodynamics of Solids under Pressure 11

isothermal (κT ) and adiabatic (κS) compressibilities: 1 ∂V 1 ∂2G κT = − = − 2 , (1.24) V ∂p T V ∂p T 1 ∂V 1 ∂2H κS = − = − 2 , (1.25) V ∂p S V ∂p S with units of pressure−1. The inverse of the compressibility, called bulk modulus, is often employed: Copyrighted 1 ∂p ∂2F BT = = −V = V 2 , (1.26) κT ∂V T ∂V T 1 ∂p ∂2U BS = = −V = V 2 . (1.27) κS ∂V S ∂V S The bulk moduli have units of pressure. These intensive properties reflect the elastic response of a solid to hydrostatic compression. The bulk modulus is one of the basic quantitiesMaterial that determines the elasticity of a material (in the case of an isotropic material, together with the Poisson ratio, although one can use equivalently other measures such as the Young modulus or the shear modulus). Typical values for the bulk modulus at ambient conditions [19, 20] are: lithium (11 GPa), NaCl (24 GPa), aluminum (76 GPa), silicon (99 GPa), steel (160 GPa), and diamond (443 GPa).- The difference between BT and BS is, in general, small compared to the value ofTaylor the bulk modulus, so the terms are often used interchangeably. The constant pressure (Cp) and constant volume (Cv) heat capacities are defined in terms of the heat necessary to raise reversibly the temperature of a closed system, and are defined as: & δQ ∂H ∂S ∂2GFrancis Cp = = = T = −T 2 , (1.28) δT p ∂T p ∂T p ∂T p Downloaded by [Taylor and Francis/CRC Press] at 12:53 06 April 2016 δQ ∂U ∂S ∂2F Cv = = = T = −T 2 , (1.29) δT V ∂T V ∂T V ∂T V

where we have used ∆H = Qp and ∆U = Qv in a closed system with pressure- volume work only undergoing constant pressure and constant volume processes, respectively. Heat capacities are extensive properties. Their equivalent intensive properties for pure substances, obtained by dividing the heat capacity by the amount of substance, are the constant pressure and constant volume molar heat capacities, usually represented in lower case (cp and cv). Cp and Cv measure a system’s ability to store energy, and in consequence they are directly related to the available microscopic degrees of freedom. In a solid, their 12 An Introduction to High-Pressure Science and Technology

temperature dependence is characteristic: Cp and Cv go to zero for T → 0 and to the Dulong–Petit limit (3R = 3NAkB per mole of atoms, with R the gas constant, NA the Avogadro’s constant, and kB the Boltzmann’s constant, 1.3806488 × 10−23 J/K) or slightly above it in the high-temperature limit. The (volumetric) coefficient of thermal expansion or thermal expansivity measures the response of volume to changes in temperature. It is defined as: 1 ∂V 1 ∂2G α = α = = . (1.30) V V ∂T V ∂T ∂p Copyrighted p The thermal expansion coefficient is a quantity with great relevance in engineering and architecture. The vast majority of materials have positive α, that is, they expand when heated at constant pressure, but some materials present extensive temperature ranges with negative thermal expansivity. A characteristic example of this behavior is the cubic zirconium tungstenate (ZrW2O8), which has a negative α at all temperatures in its stability range [21]. Likewise, some composite materials like invar, a nickel-iron alloy, have very low thermal expansion coefficientsMaterial and they are used in construction as well as in precision instruments. The equations presented above can be used to derive thermodynamic relations between these material properties. For instance, the heat capacities are related by: α2VT Cp − Cv = - . (1.31) κTaylorT Likewise, the difference between compressibilities is: α2TV κT − κS = , (1.32) Cp & and the ratio between heat capacities is the same as betweenFrancis bulk moduli and compressibilities: Cp BS κT

Downloaded by [Taylor and Francis/CRC Press] at 12:53 06 April 2016 = = . (1.33) Cv BT κS An important quantity that will appear later in this chapter is the thermal Grüneisen parameter: V αBT V αBS γth = = , (1.34) Cv Cp which represents the change in the vibrational characteristics caused by a change in volume. Because the product of the thermal coeﬃcient with the bulk modulus is a simple derivative: ∂S ∂p αBT = = , (1.35) ∂V T ∂T V Thermodynamics of Solids under Pressure 13

the Grüneisen parameter can be rewritten as: V ∂S V ∂p γth = = . (1.36) Cv ∂V T Cv ∂T V More relations, and the proofs for the ones given above can be found in Wal- lace’s book [16]. Finally, the Nernst–Simon statement of the third law of thermodynamics says that for any isothermal process that involves only substances in internal equilibrium, the entropy change ∆S goes to zero as T goes to zero. This statementCopyrighted emerged from the works by Richards, Nernst and Simon in the ﬁrst decade of the past century. Based on the arbitrary choice that the entropy of each element is zero at 0 K, the Nernst–Simon statement of the third law is used to ﬁnd conventional entropies of compounds: S0 = 0 for each element or compound in internal equilibrium, where the zero subscript represents 0 K conditions [15]. Microscopically, the entropy is related to the number of microscopic states available to an isolated system for a given volume, internal energy, and number of particles. The relation betweenMaterial entropy and the available number of states compatible with the volume, energy, and number of particles of an isolated system Ω is given by Boltzmann’s postulate:

S = kB ln Ω(U, V, N). (1.37) At 0 K, only the system’s lowest available- energy level is populated and Ω becomes the degeneracy of this energy level.Taylor For this statistical-mechanical result to be consistent to the thermodynamic result S0 = 0 for a substance, the degeneracy of the ground level of the one-component system would have to be 1. To make the statistical-mechanical and thermodynamic entropies agree, the convention of ignoring contributions to S coming from& nuclear spins and isotopic mixing is adopted (see more details in Section 21.9Francis of Reference 15). The Helmholtz free energy and the Gibbs free energy are related to diﬀerent statistical ensembles whose macroscopic states are ﬁxed by the values of their Downloaded by [Taylor and Francis/CRC Press] at 12:53 06 April 2016 natural variables [17]. These are the canonical and the isothermal-isobaric ensembles, respectively. F and G are related to the partition functions of the corresponding ensemble by a simple relation:

F (N,V,T ) = −kBT ln Z(N,V,T ), (1.38)

G(N, p, T ) = −kBT ln ∆(N, p, T ). (1.39) The partition functions are calculated using the available energy levels of the system. For instance, the canonical partition function is: X Ei(N,V ) Z(N,V,T ) = exp − , (1.40) k T i B 14 An Introduction to High-Pressure Science and Technology

where the i sums over all states available to the system for the given volume and number of particles, Ei is the energy of that state, and T is the thermodynamic temperature. For a given temperature, if the energy levels are separated by differences much larger than kBT , only the ground energy level (E0) con- tributes to the sum, and F (N,V,T ) = E0 for a non-degenerate ground energy level. Conversely, if the separation of all available energy levels is very small compared to kBT , the partition function reduces to the number of available states and F is a constant times kBT , a result known as the equipartition theoremCopyrighted in classical statistical mechanics. 1.2.2 Principle of Minimum Energy The second law of thermodynamics—that thermodynamic equilibrium in an isolated system is achieved when the entropy of the system is maximized— can be recast in several ways, depending on the independent variables we use. For every one of the thermodynamic potentials defined above (U, H, F , and G), and for a closed system (with pressure-volume work only) in which their corresponding independentMaterial variables are fixed, thermodynamic equilibrium is reached when the value of the corresponding potential is minimized. A useful generalization of the Gibbs free energy is the availability or non-equilibrium Gibbs energy of the system, G?, a function that gives the Gibbs energy of the system away from equilibrium, either material—phase and chemical—or thermal and mechanical- equilibrium [22, 23]. It can be shown from the second law that at a fixed environmentalTaylor temperature TE and pressure PE, the most stable state of the system is that at which the function ? G = U +PEV −TES has its lowest possible value. This function is only equal to the Gibbs free energy G when the system is at thermal and mechanical equilibrium with the environment (TE = T , PE = p). In& the remaining chapters of this book, for simplicity, no distinction will be made between p and PE or between G and G?, even for systems out of mechanical equilibrium.Francis Which of those quantities is used will be clear from the context. Downloaded by [Taylor and Francis/CRC Press] at 12:53 06 April 2016 Let us consider now the particular case of a one-component crystal phase. For a solid, the more convenient variables, in the sense that they can be easily controlled experimentally, are pressure and temperature. In an infinite periodic solid, there are only two sources of energy levels that enter the isothermal- isobaric partition function: electronic and vibrational. For the majority of solids at the usual temperatures, the excited electronic levels are inaccessible, so the thermal contributions to G? are mostly determined by the vibrational levels. If a system is held at a fixed temperature TE = T and a constant hydrostatic pressure PE (or simply P ), the equilibrium state minimizes the Thermodynamics of Solids under Pressure 15

non-equilibrium Gibbs energy of the crystal phase,

G?(x,V ; P,T ) = F ?(x,V ; T ) + PV ? = Esta(x,V ) + Fvib(x,V ; T ) + PV , (1.41)

with respect to all internal configuration parameters. In this equation, Esta is the electronic (static) energy of the solid that is obtainable, for instance, ? through computational techniques. Fvib is the non-equilibrium vibrational Helmholtz free energy. The structure of a periodic infinite crystal is determined by itsCopyrighted volume V and a number of coordinates, which include the variables that determine the geometry of the unit cell (angles and lattice parameters) as well as the crystallographic coordinates of the atoms whose values are not fixed by symmetry. All these variables are gathered in the x vector. Additional free energy terms can be used for other degrees of freedom that may represent accessible energy levels at the given temperature and that may contribute to the free energy of the solid, for instance, electronic contributions (in a metal), configurational entropy (in a disordered crystal), etc. The principle of minimumMaterial energy for G? says that thermodynamic equilibrium is reached at the x and V values for which G? is a minimum at given T and P . That is:

G(p = P,T ) = min G?(x,V ; P,T ). (1.42) x,V - At the x and V for which G? is a minimum,Taylor the value of the system pressure p is equal to the environmental pressure P .

1.2.3 Hydrostatic Pressure and Thermal Pressure The minimum energy principle represented by Equation& (1.42) can be used to predict x(p, T ), the volume, V (p, T ), and the equilibriumFrancis Gibbs function G(p, T ) of a crystal at any given pressure and temperature. In the context

Downloaded by [Taylor and Francis/CRC Press] at 12:53 06 April 2016 of the computational determination of these quantities, however, ﬁnding the global minimum of G?(x,V ; P,T ), which involves ﬁnding all terms in Equa- tion (1.41) for a large number of x and V values, is prohibitive. A common approach to simplify this problem is to restrict the variables x to those resulting from a minimization of the electronic (static) energy at any given volume:

xopt(V ) from Esta(V ) = min Esta(x,V ). (1.43) x

? If no crystal vibrations were present, then Fvib would disappear, and this approach would be strictly correct. Note, however, that E(x,V ) is still a huge 16 An Introduction to High-Pressure Science and Technology

potential energy surface where all crystal phases compatible with that volume are represented. Despite this, crystal structure prediction or a judicious selection of candidate phases can render this problem tractable (see Chap- ter 4). In the presence of crystal vibrations, Equation (1.43) embodies the ? approximation that the variation of Fvib with x is negligible compared to that of Esta. This “statically constrained” approximation [24–27] transforms Equation (1.41) into:

? ? G (V ; P,T ) = Esta(xopt(V ),V ) + F (xopt(V ),V ; T ) + PV , (1.44) Copyrighted vib where all degrees of freedom except for the volume have been eliminated (c.f. Equation (1.41)). The minimum energy principle can now be applied to Equation (1.44). The equilibrium volume at P and T is found by making the volume derivative of G? zero, which leads to the mechanical equilibrium condition: ∂G? = 0 = −p − p + P . (1.45) ∂V sta th MaterialP,T ? In this equation, psta = −dEsta/dV is the static pressure, pth = −(∂Fvib/∂V )T is the thermal pressure, and P is the applied external pressure. At mechanical equilibrium, the sum of static and thermal pressure balances the external pressure applied on the solid: - P = psta + pTaylorth. (1.46) The thermal pressure, which is usually negative, represents the expansion effect caused by the atomic vibrations in the crystal. It can be used, for instance, to estimate the thermal expansion of molecular crystals in order to compare static DFT with experimental x-ray crystal structures in& molecular solids [28]. If the equilibrium is stable (or metastable) then G? mustFrancis be a minimum with respect to variations (or small variations) in the volume at constant P ?

Downloaded by [Taylor and Francis/CRC Press] at 12:53 06 April 2016 and T . In other words, the second derivative of G with respect to volume (at constant T and P ) must be positive. This second derivative is equal to BT V , thus obtaining the usual criterion for mechanical stability (BT > 0). In the statically constrained approximation, the thermal pressure depends only on volume and temperature. In reality, pth depends on all variables x and particularly on the shape of the unit cell for a particular volume. The statically constrained approximation can be partially relaxed [26, 27, 29] by considering both volume and strain in Equation (1.43). The thermal pressure is then generalized to a thermal stress:

? 1 ∂Fvib(xopt(εij,V ), εij,V ; T ) σth,ij(εij,V ; T ) = , (1.47) V ∂εij Thermodynamics of Solids under Pressure 17

2500 liq. 2000 γ 1500

Temperature (K) 1000 α β

Copyrighted 500 0 5 10 15 20 25 30 35 Pressure (GPa)

Figure 1.1 Pressure-temperature phase diagram for a one-component system, show- ing three solid phases (α, β, and γ) and the liquid phase.

where ε is the strain tensor—a measure of the deformation of the solid with respect to the equilibriumMaterial geometry at volume V (also see Chapter 2). In the statically constrained approximation, the thermal stress tensor σ would be diagonal and its elements equal to minus the thermal pressure. However, even this generalization leads to a computationally prohibitive iterative process involving an “excessive number of calculations” [26]. - 1.2.4 Phase Equilibria Taylor At a given temperature T and pressure P , there are many diﬀerent atomic ar- rangements of a crystal (corresponding to diﬀerent values of V and x in Equa- tion (1.41)) that are local minima of the non-equilibrium Gibbs function G?.A corollary of the minimum energy principle is that of all these& minima, only the one with the lowest G? (the global minimum) represents theFrancis thermodynamically stable phase, and the rest are metastable phases. The two-dimensional

Downloaded by [Taylor and Francis/CRC Press] at 12:53 06 April 2016 plot that represents the stable crystal phase as a function of temperature T and pressure p is called a (pressure-temperature) phase diagram. An example is shown in Figure 1.1. The phase diagram of a solid determines the behavior of the material at arbitrary temperature and pressure conditions, and is therefore of great technological relevance. Even though only one of the phases is thermodynamically stable at a given pressure and temperature, metastable phases of pure crys- talline solids, which are not usually represented in a phase diagram, are the rule rather than the exception. Carbon is the traditional example: graphite is the stable polymorph at zero pressure and room temperature, but diamonds are observed (metastable) because the transition from diamond to graphite 18 An Introduction to High-Pressure Science and Technology

is kinetically very unfavorable under room conditions (the energy barrier the material needs to overcome to go from one phase to the other is too high, see Chapter 2). As a matter of fact, the artiﬁcial generation of diamonds requires pressures and temperatures well beyond the thermodynamic limits of the graphite phase. The boundaries between regions associated to diﬀerent phases represent particular states of a system in which two phases may coexist. For a given temperature, the pressure at which one phase transforms reversibly into an- other is called the transition pressure, ptr(T ). Along the two-phase equilibrium curves,Copyrighted the value of G(p, T ) per mole is the same for both phases. By using this result, one can calculate properties of the curve such as its slope at a given point, which is given by the Clapeyron equation: dp ∆S ∆H tr = tr = tr , (1.48) dT ∆Vtr T ∆Vtr

where ∆Str and ∆Vtr are the differences in entropy and volume between both phases at that point. DuringMaterial most phase transitions, energy is absorbed or released by the system and the volume increases or decreases without change in temperature and pressure. This “latent heat” (or the volume change) is used to drive the transition from one phase to the other. The latent heat is given by ∆Htr. From a thermodynamic point of view,- phase transitions can be classified according to the derivatives of G that are discontinuousTaylor across the transforma- tion [30]. First order phase transitions involve latent heat and are characterized by an infinite heat capacity at the transition temperature and pressure as well as by discontinuities in the properties that are derivatives of the Gibbs function: V and −S. Most phase transitions belong in this category;& boiling water is an example of a first order phase transition. In contrast, high order or continuous phase transitionsFrancis do not involve latent heat. In second-order phase transitions, there is a discontinuity in the Downloaded by [Taylor and Francis/CRC Press] at 12:53 06 April 2016 heat capacity, but it does not diverge. The isothermal compressibility and the thermal expansion coefficient, which contain also second derivatives of G with respect to T and/or p, are discontinuous too. The first derivatives of G, volume and minus entropy, show changes in their slopes, but they are continuous across the transition. In consequence, the Clapeyron equation cannot be applied. An example of a second-order phase transition is the onset of a superconducing regime in a material. See more details in Chapter 3 and also in References 15 and 30. Other phase transition classifications using different criteria are presented in Chapter 2. Thermodynamics of Solids under Pressure 19

1.3 EQUATIONS OF STATE An equation of state (EOS) is a relationship between the thermodynamic variables relevant for a given mass of a solid phase under study at equilibrium. For a solid phase under pressure, the volumetric EOS is written in terms of pressure p, volume V , and temperature T :

F (p, V, T ) = 0. (1.49) ACopyrighted simple equation of state, presented in Murnaghan’s historical paper [31], is based on the principle of conservation of mass combined with Hooke’s law for an inﬁnitesimal variation of stress in a solid. The Murnaghan EOS can also be obtained by assuming a linear variation of the isothermal bulk modulus with 0 pressure, BT (p) = B0 + B0p, and integrating BT = −V (∂p/∂V )T :

1 0 0 − /B0 B0 V (p) = V0 1 + p , (1.50) B0 which can be inverted to: Material

" 0 # B0 B0 V0 p(V ) = 0 − 1 . (1.51) B0 V

The hydrostatic work equation, δW = −-pdV , can then be used to integrate this equation of state to give: Taylor

B0 B V (V /V ) 0 B V E(V ) = E + 0 0 + 1 − 0 0 , (1.52) 0 B0 B0 − 1 B0 − 1 0 0 &0 for the volume-dependent energy at zero temperature. In these and the subsequent equations in this section, the subscript “0” representsFrancis zero-pressure conditions. That is, V0 is the (zero-pressure) equilibrium volume, E0 is the energy Downloaded by [Taylor and Francis/CRC Press] at 12:53 06 April 2016 at that volume, etc. In the rest of this section, we consider only static energies, although the same equations of state (or rather, their derivatives) can be used to fit experimental pressure-volume isotherms, which contain zero-point and temperature effects. A collection of equations of state for solids can be obtained from the strain- stress (StSs) family of relations. We describe the energy of a crystal as a Taylor expansion in the strain f: n X k E = ckf . (1.53) k=0 Some of the equations of state arising from different strain definitions follow. 20 An Introduction to High-Pressure Science and Technology

1.3.1 Birch–Murnaghan Family The Birch–Murnaghan (BM) EOS family is based on the Eulerian strain, which is deﬁned as: 1 f = (V /V )2/3 − 1 , (1.54) 2 r

where Vr, the reference volume, is usually the zero-pressure equilibrium volume of the phase under study (V0). The StSs series can be expanded to any order n > 1 (Equation (1.53)). TheCopyrighted resulting n-degree polynomial EOS has coeﬃcients that can be found by imposing the limiting conditions for the derivatives at the reference point (Vr): dE dp dB lim V ; E; p = − ; B = −V ; B0 = ; ... f→0 dV dV dp 0 = {V0; E0; 0; B0; B0; ... } . (1.55) For instance, solving these equations for the fourth-order (n = 4) Birch– Murnaghan EOS gives: Material

Vr = V0; (1.56)

c0 = E0; (1.57)

c1 = 0; (1.58) 9 c = V B ; - (1.59) 2 2 0 0 Taylor 9 c = V B (B0 − 4); (1.60) 3 2 0 0 0 3 c = V B {9[B B00 + (B0 )2] − 63B0 + 143}. (1.61) 4 8 0 0 0 0 0 0 & The polynomial strain EOS can be rearranged to produceFrancis the second-order Birch–Murnaghan equations, which are non-linear in the volume, and are typ-

Downloaded by [Taylor and Francis/CRC Press] at 12:53 06 April 2016 ically used in solid-state simulations [32]: 9 9 E = E + B V f 2 = E + B V (x−2/3 − 1)2, (1.62) 0 2 0 0 0 8 0 0 3 p = 3B f(1 + 2f)5/2 = B x−7/3 − x−5/3 , (1.63) 0 2 0 5/2 B = B0(7f + 1)(2f + 1) , (1.64)

with x = (V/V0). Higher order BM EOS result in even more cumbersome non-linear expres- sions that are not always easy to converge. Furthermore, the Eulerian strain is just one of an inﬁnite number of possible deﬁnitions [33]. Other options are CHAPTER 1 Thermodynamics of Solids under Pressure AlbertoCopyrighted Otero de la Roza MALTA Consolider Team and National Institute for Nanotechnology, National Research Council of Canada, Edmonton, Canada Víctor Luaña MALTA Consolider Team and Departamento de Química Física y Analítica, Universidad de Oviedo, Oviedo, Spain Manuel Flórez MALTA Consolider Team and Departamento de Química Física y Analítica, Universidad de Oviedo, Oviedo, Spain Material

CONTENTS

1.1 INTRODUCTION TO HIGH-PRESSURE SCIENCE

igh pressure research [1] is a ﬁeld of enormous relevance both for its sci- H entiﬁc interest and for its industrial and technological applications. Many

3 4 An Introduction to High-Pressure Science and Technology